brownian transport ii: constrained geometries · brownian transport ii: constrained geometries •...
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Brownian transport II: Brownian transport II: Constrained GeometriesConstrained Geometries
• bio-systems, porous media
• artificial submicron devices
• noise rectification mechanisms
P Hänggi and FM Rev Mod Phys 81 (2009) 387P. Hänggi and FM, Rev Mod Phys, 81 (2009) 387
Entropic channelsEntropic channels
)(teFr x ξrrr
& +=Augsburg group, PRL 96, 2006, PRE75, 2008
ξ(t): gaussian, ‹ξι(t)›=0‹ξi(t)ξj(0)›=2D0δijδ(t)
w(x): channel radius
[ ] )(/)()()( xPDxAxDxP ′+∂∂∂Zwanzig, JCP, 1992 [ ] )(/)()()( 0 xPDxAxDxP xxt +∂∂=∂
entropic term)(ln0 xDFx σ−−=)(xAD
σ(x)=2w(x) in 2D, =πw(x)2 in 3D, channel cross section
entropic term)(ln0 xDFx σ)(xAD0
D(x)=D0/[1+w’(x)2]α, α =1/2 (3D), 1/3 (2D)
Reguera, Rubi, PRE64, 2001
A(x)
mobilitymobility
tt
FxF
/)]0()([li
/)( =
&
&μ
txtxxt
/)]0()([lim −=∞→
μ(F) → 1, for F → ∞ theory fails
diff i itdiff i itBurada et al, ChemPhysChem 10, 2009
diffusivitydiffusivity
ttxtxD 2/])()([lim 22 −= ttxtxDt
2/])()([lim=∞→
D(F) goes through a depinning peak,and D(F) → D for F → ∞and D(F) → D0 for F → ∞
Costantini, FM, EPL 48, 1999
Mobility and Diffusion in Mobility and Diffusion in a periodic potentiala periodic potentiala periodic potentiala periodic potential
Model: single Brownian particle in a tilted washboard potential
L=2πwashboard potential
FtxVxxm ++′−−= )()( ξγ &&& μ=v/F v∞=F/γ
γ= 0.01
kT=1
ξ(t): thermal noise ⟨ξ⟩ = 0, ⟨ξ(t)ξ(0)⟩ = 2γ kTδ(t)V(x)=V(x+L): periodic substrate
γ = 0.1
validityvalidityyy
1 fast transverse re-equilibration
Burada et al, PRE75, 2007
1. fast transverse re-equilibration
ty << max{tx,tF}
ty =w2max/2D0 transverse diffusion
tx = L2/2D0 longitudinal diffusionP(δx) ≈ F/D0 exp(-Fδx/D0)
tF=L/F drift
2. smooth channel corrugationsdespite much effort, validity of the reductionist
P(δx) F/D0 exp( Fδx/D0)
2. smooth channel corrugations
w’(x) << max{1,D0/FL}
f l d
validity of the reductionist approach is restricted restricted to special geometries
more stringent for large drives
Laachi et al, EPL 80, 2007
BonaBona--fidefide 2D and 3D channels2D and 3D channelsBonaBona fidefide 2D and 3D channels2D and 3D channels
)(teFr x ξrrr
& +=
Fick Jacobs approach failsFick-Jacobs approach fails
even for F=0
F
r1
r
F
r0
interest not conceptual only
applicationsapplicationsapplicationsapplications
channel networks in natural andchannel networks in natural and
artificial porous media (zeolites,
membranes, etc)
particle ratcheting in 2D and 3D
channels is more effective for sharp
boundaries (eg. magnetic vortices in
type-II superconductors)
Nori, F.M., PRL 83, 1999
2D septate channels2D septate channels
structure-less pores
reflecting walls
stationary regime
P(δ ) ( Fδ /D )
mobility curvemobility curveF-1/2
P(δx) ~ exp(-Fδx/D0)
decreasesdecreases monotonicallymonotonically
F
μμ(∞) = Δ/yL independent of xL
for D0/F << Δ
(F) ( ) F 1/2 l dμμ(F) − μμ(∞) ∝ F-1/2, slow decay
μμ(0) depends on Δ/yL and xL/yL Savelev, FM, PRE 80, 2009
zerozero--drive mobilitydrive mobility
logarithmic dependence on Δ/yL
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ−=
LL
L
yxy ln21
1)0(
π
μL
LL
yyx
<<Δ>>
⎠⎝ LL y
• μ(0)=x2L/2D0τ0 with τ0 a MFPTμ( ) L/ 0 0 0
• map the FPE into a Poisson eqnfor the electrostatic field in a for the electrostatic field in a multistrip chamber (Purcell 1977)
τ0 ⇔ capacitance C0
• CERN report 77/09 for C0
diffusivitydiffusivity
NO excess diffusion peak
NO horizontal asymptote D(F)=D
23
112
)(⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ−
Δ=
DF
DFD
NO horizontal asymptote D(F)=D0
quadratic Fquadratic F dependence (Taylor’s type)00 12 ⎟
⎠⎜⎝
⎟⎠
⎜⎝ DyD L
• = v(t) is a dichotomic process v∈{0,F}
)(tx&
⎞⎛•
/
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
F
F tFtCττ
ττττ
02
02 exp)(
with τ0+τF=τ, τF /τ =Δ/yL
and τ0=(1-Δ/yL)τ = [(yL-Δ)/2]2/3D0
• ∫∞
=0
)()( dttCFD
3D septate channels3D septate channels
r1important difference: two transverse diffusion times,ty =r1
2/2D0 and MFPT out of the annulus, with r0
y 1 / 0 ,absorbing inner walls (large F regime), tr=r0
2/2D0 ln(r1/r0)
mobility curvemobility curve
μμ(∞) = (r0/r1)2
for D0/F << r0
μμ(F) − μμ(∞) ∝ F-1, faster decayy
μμ(0) easier to compute
zerozero--drive mobilitydrive mobility
linear dependence on pore radius
21
02)0(rrxL
πμ = Berezhkovskii et al, JCP 116 (2002);118 (2003)
1rπ
• μ(0)=x2L/2D0τ0μ( ) L 0 0
• Poisson eqn for a circular disk symmetrically placed between two parallel conducting planes 2xL apart.
τ0 ⇔ capacitance C0
• C0=8r0
diffusivitydiffusivity2
01
0
2
21
20
2
21
20
21
0
ln12
)(⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
DF
rr
rr
rrr
DFD
01110 2 ⎠⎝⎠⎝⎥⎦⎢⎣ ⎠⎝⎠⎝ DrrrD
quadratic in F, like in 2D
• = v(t) is a dichotomic process v∈{0,F}
)(tx&
• ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
F
F tFtCττ
ττττ
02
02 exp)(
with τ0+τF=τ and τF=[1-(r0/r1)2]τ = r0
2/2D0 ln(r1/r0)
• ∫∞
=0
)()( dttCFD
Eccentric pore geometriesEccentric pore geometries
yLxxh
xL
• no “free lane” along the x-axis
f l F D /F Δ diff i i t• for large F, D0/F << Δ, diffusion againstcompartment wall is 1D with exit time
τ≈(xh-Δ/2)2/2D0 F
mobilitymobility μ
for large F
v=xL/τ being τ>>xL/F
FxF L
F τμ ⎯⎯ →⎯ ∞→)(
diffusivitydiffusivity
(F)/D
0
yysame as for a driven
free Brownian walker D
(
2
22
2)(
τσ
ττ×= LxFD
F/D0
… for a better estimate Cox formula
Absolute Negative MobilityAbsolute Negative Mobility
•• special channel geometryspecial channel geometry
•• spherical, pointspherical, point--like particleslike particles
•• F F →→ FF00 + F+ F11cos (cos (ΩΩt) t) with F0 << F1
F0
jin the presence
of an ac drive j= x& j
x
of an ac drive j
flows against the
static bias F0
x
F0 y0
Reimann, Hanggi, PRL88, 2002
ANM in a septate channelANM in a septate channelF → F(t) = F0 + F1(t) with F1>F0
F1+F0
(F F )t
v(F)=μ(F) F
-(F1-F0)
v(F) μ(F) F
μ(F) monotonically decreasing function
Q:Q: can v(F1+F0) < v(F1-F0)?ANM?ANM?
QQ ( 1 0) ( 1 0)
A:A: no ANM for symmetric particles as μ(F) decays not faster than 1/F!
Recipe:Recipe: We must “lower” the symmetry of the system!
pore asymmetry →Arrhenius traps
back to the original schemeEichhorn et al, PRL88, 2002 trap δxtδxt
particle asymmetry →ellipsoidal objects ellipsoidal objects
ANM predictedANM predicted
ANM for (ANM for (prolateprolate) ellipsoidal particles: ) ellipsoidal particles: δδxxtt = b= b--aa
exit time longer for exit time longer for F1 || F0
F F ⎟⎞
⎜⎛ −± 0)()( FabFe.g. F0 < F1 ⎟⎟
⎠⎜⎜⎝±=±
0
010
)(exp)(D
Fττ
optimal ANM for
⎟⎞
⎜⎛ −⎟
⎞⎜⎛ − )()( FabFab
⎟⎟⎠
⎞⎜⎜⎝
⎛+<Ω<⎟⎟
⎠
⎞⎜⎜⎝
⎛−
0
00
0
0 )(exp)(expD
FabD
Fab τJ
sensitivity to other parameters
eccentricity D etceccentricity, D0, etc
ConclusionsConclusions
• channels in 2D and 3D show properties that are unaccounted p p
for in Fick-Jacobs approximation
• septate channels as reference solution for the design of
artificial nanodevices. Next: extended particles, chains, etc
• I wish to thank my coworkers atAugsburg University, Germany (P. Hänggi, G. Schmid, P.S. Burada)
Loughborough University, UK (S. Savelev, A. Pototskii)
Perugia University, Italy (M. Borromeo)
RIKEN, Japan (F. Nori, P. Ghosh)